CN109543234A - A kind of component life estimation of distribution parameters method based on random SEM algorithm - Google Patents
A kind of component life estimation of distribution parameters method based on random SEM algorithm Download PDFInfo
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Abstract
The invention belongs to be specially adapted for the equipment of the numerical calculation of specific application or data processing or method and technology field, a kind of component life estimation of distribution parameters method based on random SEM algorithm is disclosed；The survival function of lifetime of system, and the maximum likelihood function in write through system service life are found out with the cumulative distribution function of component life and system signature vector；Estimated using random SEM algorithm parameters of operating part: S step is combined with order system signature and conditional probability vector and obtains life estimate sample, and M step brings estimation sample into likelihood function and maximizes undated parameter；System small sample and in carrying out Monte Carlo simulation experiment under two kinds of situations of large sample and generate the deviation and mean square error, coverage rate, confidence interval expected width four indices of point estimation；It finds that the estimation of SEM algorithm is more excellent compared with previous algorithm, loses system estimation component life distribution parameter situation for handling bilateral delete of II type.
Description
Technical field
The invention belongs to be specially adapted for the equipment of the numerical calculation of specific application or data processing or method and technology field,
A kind of more particularly to component life estimation of distribution parameters method based on random SEM algorithm.
Background technique
Currently, the prior art commonly used in the trade was such that in 1985, the concept and its basic calculating of signature
Mode was proposed that since then, signature vector starts to be attempted addition system reliability by Samaniego for the first time in 1985
Within the Method and kit for of research；Later in 1999, Kocha et al. develops system lifetim distribution function can be by
The mixing of signature vector and component life distribution function is expressed, and further established contacting between vector and system；Separately
Outside, Boland provided the interconnected system Signature calculation method based on system road collection number in 2001.By working above,
The concept of system Signature is profoundly applied in each branch of system Reliability Research extensively by people.But meanwhile by
In the existing limitation (calculated lifetime of system structure function is often more complicated) based on signature method, solving
Many time is consumed in the process, is unfavorable for the estimation and calculating of parameter, therefore the parameter for becoming the distribution of system unit service life is estimated
A hot and difficult issue problem in meter.The key link that the simultaneity factor service life is predicted is estimated to unknown parameter
Meter decides predicting whether accurately for lifetime of system.Up to now scholar has explored several for the system unit service life point
Cloth carries out the algorithm of parameter Estimation, such as: for the unilateral minimum dispersion linear unbiased estimator algorithm for deleting mistake of system, optimum linearity is unbiased
Algorithm for estimating, moments estimation algorithm, bayesian algorithm, maximum likelihood estimation algorithm, based on regression estimation algorithm etc..The prior art
The distribution of component life is discussed in one " method for parameter estimation being distributed based on minimum dispersion linear unbiased estimator to the system unit service life "
Infer with linear, which is derived in general scale parameter Zu Ban scale parameter race and Location-Scale Parameter race parameter most
Excellent linear unbiased estimate, and give the accurate calculation formula of variance and covariance.But this method only considered a kind of component longevity
Distribution parameter is ordered, does not take into account more parameter factors, this causes the method uncertain for component life family of distributions
System be difficult to be estimated.Based on the prior art two " method for parameter estimation being distributed by moments estimation to the system unit service life "
It is relatively simple to calculate Method And Principle, does not have cumbersome calculating process compared to other algorithms, but this but also only sample size compared with
Accurate estimation could be obtained when big.In practical applications, it will lead to estimated result in the case of sample size is less
Deviation it is larger.The prior art three " method for parameter estimation that the system unit service life is distributed based on Bayesian Estimation ", this method
Solve the parameter Estimation of non-monotonic crash rate distribution situation, but this method relies heavily on the selection of loss function,
Can have a great impact to the Optimality of estimator if loss function does not have good describing system, the technology is in asymmetric damage
It is difficult to obtain the higher estimated result of precision in the case of mistake.The prior art four " is distributed the system unit service life based on maximum likelihood
Method for parameter estimation " with order statistic very good solution Random censorship the problem of, but theoretical local derviation is tediously long, Er Qiexu
Numerical Methods Solve is wanted, the convergence of calculated result cannot be guaranteed, and often will appear the case where result does not restrain.Existing skill
Art five " based on the method for parameter estimation being distributed to the system unit service life is returned ", the advantages of this method be it is relatively simple understandable, no
Operation time is also saved while needing additional calculation process, then provide a desired result, and can be used as its other party
The initial value of method brings calculating into.But this method is also required to numerical solution, and the convergence of calculating cannot be completely secured, when missing data ratio
When example is very high, obtained result is often unstable.
In conclusion problem of the existing technology is: the method for parameter estimation in existing system unit service life, which exists, to be borrowed
It helps and needs the numerical problem of huge calculation amount or calculating process is cumbersome and the convergence of Numerical Methods Solve cannot be guaranteed
Second order local derviation problem, this leads to have the calculating process the more demanding of higher time complexity and hardware, eventually leads to
Operation time is tediously long and precision is not high, is unfavorable for applying on to time and the higher engineering of required precision；In addition, existing
Spininess is in the majority for the unilateral system for deleting mistake in technology, and considers that II type bilateral the case where deleting mistake is less, this causes not having nowadays
There is a kind of suitable method to delete mistake system for bilateral, but regardless of being had much in the engineering fields such as aerospace or building
The bilateral system for deleting mistake, the shortcoming of research and the blank of technology develop slowly certain fields.Original method scope of application
Have a certain limitation, concrete embodiment be only used in engineering it is of less demanding to the time, it is not high to required precision unilateral to delete mistake
Model.
Solve the difficulty and meaning of above-mentioned technical problem: in the parameter Estimation being distributed to the system unit service life, if occurring
System unit service life unknown situation, previous method are difficult to be calculated, if bringing general component life distribution into, can lead
Causing the levels of precision of estimation reduces.If when occur evaluation solution result is tediously long and second order local derviation be unable to get it is convergent
Situation, existing technology are just difficult to carry out, and lead to the narrow scope of application of signature method.In addition, double there is II type
While deleting mistake situation, the estimation difficulty that the missing of bilateral data will lead to parameters of operating part is increased, and the levels of precision of estimation also can be therewith
It reduces.The Optimality of the parameter Estimation of component life distribution influences whether the precision of prediction to lifetime of system；Only find one kind
It is applied widely, the parameter distribution to component life can accurately be estimated, lifetime of system could be made accurately pre-
It surveys.
Summary of the invention
In view of the problems of the existing technology, the component life distribution based on random SEM algorithm that the present invention provides a kind of
Method for parameter estimation.
The invention is realized in this way a kind of component life estimation of distribution parameters method based on random SEM algorithm, described
Component life estimation of distribution parameters method based on random SEM algorithm includes:
(1) survival function of lifetime of system is found out with the cumulative distribution function of component life and system signature vector；
(2) parameters of operating part is estimated using random SEM algorithm, S step is general with order system signature and condition
Rate vector, which combines, obtains life estimate sample, and M step brings estimation sample into likelihood function and maximizes undated parameter；
(3) it simultaneously, generates point to progress Monte Carlo simulation experiment under two kinds of situations of large sample in system small sample and estimates
The deviation and mean square error of meter, coverage rate, width four indices expected from confidence interval.
Further, the component life estimation of distribution parameters method based on random SEM algorithm specifically includes the following steps:
Step 1, according to System structural function, computing system signature vector SΓ(s1,s2...sn) and order
Signature vectorI-th thrashing is arranged with ω, it is concluded that system
I-th of component of signature vector are as follows:
Event A is that k-th of order thrashing is caused by i-th of order component failure；One determining system is produced
Raw m lifetime of system, and the failure for observing which component causes thrashing, repeats this process n times；Then order system
Signature vector can be approximated to be:
Step 2, with the cumulative distribution function F of component lifeXWith system signature vector SΓIndicate lifetime of system
Survival functionTo n independent same distribution component system, the stochastic variable of component life is indicated with X, system is indicated with T
The stochastic variable in service life；The then survival function of lifetime of systemIt indicates are as follows:
Step 3 is established random SEM algorithm and is estimated parameters of operating part；S step, with order system signature and item
Part probability vector combines and obtains life estimate sample；M step brings estimation sample into likelihood function and maximizes undated parameter；Weight
Multiple S step and M step, iteration obtain parameters of operating part estimated value；
Bilateral 100 (1- α) % confidence intervals of step 4, construction confidence interval, component distribution parameter μ and σ can construct such as
Under:
Further, the step 3 specifically includes:
(1) right truncation density formula and left truncation density formula first can be write out according to the sigma-t of order component life
It is respectively as follows:
(2) for r-l+1 system of first to r-th failure in m system；
(3) likelihood function is maximized to obtain θ relative to θ(h+1)It is recycled next time:
(4) step (2) are repeated, (3) step obtains θ(h), the sequence of h=1,2 ..., B, several iterative values before abandoning, from remaining
Iterative value in be averaged to obtain the estimation of θ
Further, the r-l+1 system for first to r-th failure in m system, component life is by such as
Lower method generates:
(1) k-th of system (being sorted based on lifetime of system), based on order system signature, with probability mass functionλ=1,2 ..., n generate a discrete random variable Λ, this realization is expressed as λ；
(2) it is distributed from the condition of left truncation density, enables θ=θ(h)Generate λ -1 stochastic variables:
(3) it is distributed from the condition of right truncation density, enables θ=θ(h)Generate n- λ stochastic variable:
(4) the pseudo- Complete Sample of system k is finally obtained
(5) whereinTo k=l ..., r repeats step (1)-(3), obtains puppet Complete Sample (r-l+1) n to m
Each system of preceding l-1 and rear m-r system in a system.
Further, described to obtain a each system of preceding l-1 in m system of puppet Complete Sample (r-l+1) n and rear m-r
System, component life generate by the following method:
(1) it is based on conditional probability vector p(j:m), with probability mass functionδ=0,1 ..., n-1 are produced
A raw discrete random variable Δ, it is known that τ=tr:m；
(2) it is distributed from the condition of left truncation density with point of cut-off tk=tr:m, θ=θ(h)Generate δ stochastic variable
(3) it is distributed from right truncation density conditions with point of cut-off tk=tr:m, θ=θ(h)Generate n- δ stochastic variable
(4) (component life) pseudo- Complete Sample of j-th of truncation system is obtained
(5) step (1)-(4) are repeated and obtain estimation sampleL-1, r+1, r+2 ..., m
Another object of the present invention is to provide the component life distribution parameters based on random SEM algorithm described in a kind of application
The II type of estimation method is bilateral to delete mistake system.
Another object of the present invention is to provide the component life distribution parameters based on random SEM algorithm described in a kind of application
The Vehicle Engineering component life estimation of distribution parameters system of estimation method.
Another object of the present invention is to provide the component life distribution parameters based on random SEM algorithm described in a kind of application
The material engineering component life estimation of distribution parameters system of estimation method.
Another object of the present invention is to provide the component life distribution parameters based on random SEM algorithm described in a kind of application
The electronic engineering component life estimation of distribution parameters system of estimation method.
Another object of the present invention is to provide the component life distribution parameters based on random SEM algorithm described in a kind of application
The computer of estimation method.
In conclusion advantages of the present invention and good effect are as follows: the present invention considers component life estimation of distribution parameters and exists
II type is bilateral to delete situation under losing, be in actual component life estimation of distribution parameters it is indispensable, it is several compared to other
Kind deletes mistake situation, and application range is wider.The present invention uses the estimation based on maximum likelihood, the estimation based on recurrence, based on random
Three kinds of estimation methods of SEM algorithm, and use the influence of order statistic reduction Random censorship；Three kinds of estimation methods are mutually complementary
It fills, so that the parameter Estimation of component can achieve preferable effect in point estimation and interval estimation.The present invention can be effective
Components of processing systems service life distribution parameter deletes the problem of mistake is estimated II type is bilateral.Compared with prior art, of the invention to estimate
Meter precision is higher, and the scope of application is wider.
Detailed description of the invention
Fig. 1 is the component life estimation of distribution parameters method flow provided in an embodiment of the present invention based on random SEM algorithm
Figure.
Fig. 2 is system 1 provided in an embodiment of the present invention in system number m=10, is deleted when mistake rate is q=0%-50% with random
Four item data figures of SEM method emulation.
Fig. 3 is system 2 provided in an embodiment of the present invention in system number m=10, is deleted when mistake rate is q=0%-50% with random
Four item data figures of SEM method emulation.
Fig. 4 is system 3 provided in an embodiment of the present invention in system number m=10, is deleted when mistake rate is q=0%-50% with random
Four item data figures of SEM method emulation.
Fig. 5 is system 1 provided in an embodiment of the present invention in system number m=50, is deleted when mistake rate is q=0%-90% with random
Four item data figures of SEM method emulation.
Fig. 6 is system 2 provided in an embodiment of the present invention in system number m=50, is deleted when mistake rate is q=0%-90% with random
Four item data figures of SEM method emulation.
Fig. 7 is system 3 provided in an embodiment of the present invention in system number m=50, is deleted when mistake rate is q=0%-90% with random
Four item data figures of SEM method emulation.
Fig. 8 is system 2 provided in an embodiment of the present invention in system number m=10, deletes when mistake rate is q=0%-50% and uses MLE
Four item data figures of method emulation.
Fig. 9 is system 2 provided in an embodiment of the present invention in system number m=50, deletes when mistake rate is q=0%-90% and uses MLE
Four item data figures of method emulation.
Figure 10 is system 2 provided in an embodiment of the present invention in system number m=10, deletes when mistake rate is q=0%-50% and uses REG
Four item data figures of method emulation.
Figure 11 is system 2 provided in an embodiment of the present invention in system number m=50, deletes when mistake rate is q=0%-90% and uses REG
Four item data figures of method emulation.
Specific embodiment
In order to make the objectives, technical solutions, and advantages of the present invention clearer, with reference to embodiments, to the present invention
It is further elaborated.It should be appreciated that the specific embodiments described herein are merely illustrative of the present invention, it is not used to
Limit the present invention.
Exist for the method for parameter estimation in existing system unit service life by the numerical value side for needing huge calculation amount
Method or the Second Order Partial guiding method that calculating process is cumbersome and the convergence of Numerical Methods Solve cannot be guaranteed；In existing technology
Consider that II type bilateral the case where deleting mistake is less, the scope of application has the problem of certain limitation.The present invention is using the tired of component life
Product distribution function and system signature vector；Using small sample to the Monte Carlo simulation under two kinds of situations of large sample in
Experiment；More particularly to a kind of component life estimation of distribution parameters is carried out based on random SEM algorithm mistake system of deleting bilateral to II type
Method.
Application principle of the invention is explained in detail with reference to the accompanying drawing.
As shown in Figure 1, the component life estimation of distribution parameters method provided in an embodiment of the present invention based on random SEM algorithm
The following steps are included:
S101: the existence letter of lifetime of system is found out with the cumulative distribution function of component life and system signature vector
Number；
S102: estimating parameters of operating part using random SEM algorithm, S step, with order system signature and condition
Probability vector, which combines, obtains life estimate sample, and M step brings estimation sample into likelihood function and maximizes undated parameter；
S103: it simultaneously, is generated in progress Monte Carlo simulation experiment under two kinds of situations of large sample in system small sample
The deviation and mean square error of point estimation, coverage rate, width four indices expected from confidence interval.
Component life estimation of distribution parameters method provided in an embodiment of the present invention based on random SEM algorithm specifically include with
Lower step:
Step 1, according to System structural function, computing system signature vector SΓ(s1,s2...sn) and order
Signature vectorIt is assumed that i-th thrashing is arranged with ω, it is concluded that system
I-th of component of signature vector are as follows:
It is assumed that event A, which is k-th of order thrashing, to be caused by i-th of order component failure.Determining it is to one
System generates m lifetime of system, and the failure for observing which component causes thrashing, repeats this process n times.Then order system
System signature vector can be approximated to be:
Step 2, with the cumulative distribution function F of component lifeXWith system signature vector SΓIndicate lifetime of system
Survival functionTo n independent same distribution component system, the stochastic variable of component life is indicated with X, system is indicated with T
The stochastic variable in service life；The then survival function of lifetime of systemIt indicates are as follows:
Step 3 is established random SEM algorithm and is estimated parameters of operating part；S step, with order system signature and item
Part probability vector combines and obtains life estimate sample；M step brings estimation sample into likelihood function and maximizes undated parameter.Weight
Multiple S step and M step, iteration obtain parameters of operating part estimated value；
Step 4 constructs confidence interval.Bilateral 100 (1- α) % confidence intervals of component distribution parameter μ and σ can construct such as
Under:
In a preferred embodiment of the invention, step 3 specifically includes:
(1) right truncation density formula (6) and left truncation density first can be write out according to the sigma-t of order component life
Formula (7) is respectively as follows:
(2) for r-l+1 system of first to r-th failure in m system, component life produces by the following method
It is raw:
K-th of system of I (is sorted) based on lifetime of system, based on order system signature, with probability mass functionλ=1,2 ..., n generate a discrete random variable Λ, this realization is expressed as λ.
II is distributed from the condition of left truncation density, enables θ=θ(h)Generate λ -1 stochastic variables:
III is distributed from the condition of right truncation density, enables θ=θ(h)Generate n- λ stochastic variable:
IV finally obtains the pseudo- Complete Sample of system k
V is whereinTo k=l ..., r repeats step (I-III), obtains puppet Complete Sample (r-l+1) n
To in m system each system of preceding l-1 and rear m-r system, component life generate by the following method:
I is based on conditional probability vector p(j:m), with probability mass functionδ=0,1 ..., n-1 are generated
One discrete random variable Δ, it is known that τ=tr:m；
II is distributed from the condition of left truncation density with point of cut-off tk=tr:m, θ=θ(h)Generate δ stochastic variable
III is distributed from right truncation density conditions with point of cut-off tk=tr:m, θ=θ(h)Generate n- δ stochastic variable
IV obtains (component life) pseudo- Complete Sample of j-th of truncation system
V repeats step (I-IV) and obtains estimation sampleJ=1,2 ..., l-1, r+1, r+2 ..., m；
(3) likelihood function is maximized to obtain θ relative to θ(h+1)To be recycled next time:
(4) step (2) are repeated, (3) step obtains θ(h), the sequence of h=1,2 ..., B, several iterative values before abandoning, from remaining
Iterative value in be averaged to obtain the estimation of θ
Application principle of the invention is further described with reference to the accompanying drawing.
It is provided in an embodiment of the present invention to be deleted under mistake system to the system unit service life based on signature vector II type is bilateral
The method of estimation of distribution parameters includes:
(1) derivation of lifetime of system distribution function and likelihood function
The stochastic variable of component life is indicated with X, and the stochastic variable of lifetime of system is indicated with T.The cumulative distribution letter of X
Number (cdf), probability density function (pdf), survival function (sf) are expressed asThe cumulative distribution function of T
(cdf), probability density function (pdf), survival function (sf) are expressed asTo n independent same distribution component
System indicates the survival function of lifetime of system with the cumulative distribution function of component life:
Wherein siFor i-th of element of n component system signature vector, si=Pr (T=XI:n),Because
The out-of-service sequence of n independent same distribution component has n！The possible setting of kind, each setting lower component failure have specific suitable
Sequence.It is then supposed that i-th thrashing is arranged with ω, then i-th of component of the signature vector of system is finally obtained
For si=ω/n！.
By the joint density to order statistic (X (l), X (2) ..., X (r)), the hypothesis of system all parts is utilized
Integral age distribution function (weibull distribution) derives the survival function of lifetime of system:
Enabling θ=(μ σ) ' is parameter vector.It is obtained by survival function based on the bilateral θ for deleting mistake lifetime of system data of II type
Likelihood function are as follows:
In order to estimate θ, the present invention is maximized (11), or formula (12) of equal value is maximized:
(2) derivation of the sigma-t formula of order signature and condition order component
Consider the service life X of n component in k-th of system (total m system)k1, Xk2..., Xkn, k=1,2 ..., m are right
The order component life X answeredK, 1:n< XK, 2:n< ... < XK, n:nIntuitively, the system that first fails in m system it is more likely that because
For caused by critical component failure.If the arrangement of lifetime of system gives, service life TA:mAnd TB:mThe system of (a < b)
Signature vector should be different.More accurately, Balakrishnan et al. is by k-th of order lifetime of system TK:mBe
System signature vector is expressed asWherein:
Above formula is a vector, indicates that the failure of k-th of system is general due to caused by its i-th of order component failure
Rate.It enablesIndicate the quantity of the deactivation system as caused by i-th of order component failure,ThenIt can indicate are as follows:
WhereinIndicate that i-th of order component failure causes the conditional probability of k-th of thrashing, it is known thatA system
System failure is as caused by its j-th of order component failure.If lifetime of system is independent same distribution (iid),It can table
It is shown as:
(3) the sigma-t formula of order component
It is assumed that the λ component failure in k-th of system leads to thrashing, it is expressed as tk=xk,λ:nThen, remaining n-1
The condition distribution of component is the stochastic variable of right truncation or left truncation distribution.Specifically, it is known that tk=xk,λ:nPreceding λ -1 order
Component life xk,1:n,xk,2:n,…,xk,(λ-1):nSigma-t be a right truncation density i.e.:
It is also known that tk=xk,λ:nN- λ order component life x afterwardsk,(λ+1):n,xk,(λ+2):n,…,xk,n:nSigma-t
It is a left truncation density:
Conditional probability vector:
(4) principle of the estimation method based on random SEM
(i) for r-l+1 system of first to r-th failure in m system, component life produces by the following method
It is raw:
K-th of system of I (is sorted) based on lifetime of system, based on order system signature, with probability mass functionλ=1,2 ..., n generate a discrete random variable Λ, this realization is expressed as λ.
II is distributed from the condition of left truncation density, enables θ=θ(h)Generate λ -1 stochastic variables:
III is distributed from the condition of right truncation density, enables θ=θ(h)Generate n- λ stochastic variable:
IV finally obtains the pseudo- Complete Sample of system k
V is whereinTo k=l ..., r repeats step (I-III), obtains puppet Complete Sample (r-l+1) n
To in m system each system of preceding l-1 and rear m-r system, component life generate by the following method:
I is based on conditional probability vector p(j:m), with probability mass functionδ=0,1 ..., n-1 are generated
One discrete random variable Δ, it is known that τ=tr:m；
II is distributed from the condition of left truncation density with point of cut-off tk=tr:m, θ=θ(h)Generate δ stochastic variable
III is distributed from right truncation density conditions with point of cut-off tk=tr:m, θ=θ(h)Generate n- δ stochastic variable
IV obtains (component life) pseudo- Complete Sample of j-th of truncation system
V repeats step (I-IV) and obtains estimation sampleJ=1,2 ..., l-1, r+1, r+2 ..., m；
(ii) likelihood function is maximized to obtain θ relative to θ(h+1)To be recycled next time:
(iii) (i) is repeated, (ii) step obtains θ(h), the sequence of h=1,2 ..., B, several iterative values before abandoning, from remaining
Iterative value in be averaged to obtain the estimation of θ
The description of tracking problem placed m system of n component in hypothesis life test, each system
Signature is (s1,s2...sn).In life test, since being tested first of thrashing, when r-th of thrashing
When experiment terminate, wherein l, r are that subject realizes 1 determining < l < r < m.Observing obtained sequential system out-of-service time is
Tl:m< Tl+1:m< ... < Tr:mLifetime of system sample is lost to get to bilateral delete of an II type.The present invention is mainly studied by observing
The parameter being distributed to lifetime of system data to component life estimates, thus hypotheses be all life test datas
It obtains.
How to carry out accurate parameter Estimation to the component life distribution of system is the top priority to be solved.Component life
The parameter Estimation inaccuracy of distribution is mainly since observation sample will appear the case where data delete mistake.In actual life test
In, the time of all thrashings tends not to completely record, and exists beyond the life test time and still has system not
Failure causes to observe data be right truncation and deletes mistake, there is also preceding several thrashing times it is too short be difficult to record or lose, cause to see
Mistake is deleted in the left truncation of measured data.The parameter Estimation that these data for deleting mistake can be such that component life is distributed becomes inaccuracy.Thus right
In the parameter Estimation of system unit service life distribution, needs to delete the influence for losing situation in view of difference, guarantee to be distributed component life
Parameter accurately estimated.In addition, Random censorship number also can to component life be distributed parameter Estimation have an impact,
Random censorship is more, then estimates more inaccurate.On the other hand, to consider the influence that algorithm itself generates, what algorithms of different generated
Point estimation and interval estimation order of accuarcy are variant, and estimated bias is not positive and negative also identical.
Application principle of the invention is further described combined with specific embodiments below.
Assuming that the sequential system out-of-service time observed is T in life testl:m< Tl+1:m< ... < Tr:mIf directly
It connects and these data is subjected to parameter Estimation as partial data, then have ignored influence of the bilateral data for deleting mistake to parameters of operating part.
Therefore it needs to obtain pseudo- Complete Sample to the test data supplement observed.
What present example provided carries out component life distribution parameter based on random SEM algorithm mistake system of deleting bilateral to II type
The method of estimation specifically includes the following steps:
Step 1, according to System structural function, computing system signature vector SΓ(s1,s2...sn) and order
Signature vectorIt is assumed that i-th thrashing is arranged with ω, it is concluded that system
I-th of component of signature vector are as follows:
It is assumed that event A, which is k-th of order thrashing, to be caused by i-th of order component failure.Determining it is to one
System generates m lifetime of system, and the failure for observing which component causes thrashing, repeats this process n times.Then order system
System signature vector can be approximated to be:
Step 2, with the cumulative distribution function F of component lifeXWith system signature vector SΓIndicate lifetime of system
Survival functionTo n independent same distribution component system, the stochastic variable of component life is indicated with X, system is indicated with T
The stochastic variable in service life.The then survival function of lifetime of systemAre as follows:
Step 3 is established random SEM algorithm and is estimated parameters of operating part: S step, with order system signature and item
Part probability vector, which combines, obtains life estimate sample, and M step brings estimation sample into likelihood function and maximizes undated parameter.Weight
Multiple S step and M step, iteration obtain parameters of operating part estimated value
Step 4 constructs confidence interval.Bilateral 100 (1- α) % confidence intervals of component distribution parameter μ and σ can construct such as
Under
Further, step 3 specifically includes:
(1) right truncation density formula (6) and left truncation density first can be write out according to the sigma-t of order component life
Formula (7) is respectively
(2) for r-l+1 system of first to r-th failure in m system, component life produces by the following method
It is raw:
K-th of system of I (is sorted) based on lifetime of system, based on order system signature, with probability mass functionλ=1,2 ..., n generate a discrete random variable Λ, this realization is expressed as λ.
II is distributed from the condition of left truncation density, enables θ=θ(h)Generate λ -1 stochastic variables:
III is distributed from the condition of right truncation density, enables θ=θ(h)Generate n- λ stochastic variable:
IV finally obtains the pseudo- Complete Sample of system k
V is whereinTo k=l ..., r repeats step (I-III), obtains puppet Complete Sample (r-l+1) n
To in m system each system of preceding l-1 and rear m-r system, component life generate by the following method:
I is based on conditional probability vector p(j:m), with probability mass functionδ=0,1 ..., n-1 are generated
One discrete random variable Δ, it is known that τ=tr:m；
II is distributed from the condition of left truncation density with point of cut-off tk=tr:m, θ=θ(h)Generate δ stochastic variable
III is distributed from right truncation density conditions with point of cut-off tk=tr:m, θ=θ(h)Generate n- δ stochastic variable
IV obtains (component life) pseudo- Complete Sample of j-th of truncation system
V repeats step (I-IV) and obtains estimation sampleJ=1,2 ..., l-1, r+1, r+2 ..., m；
(3) likelihood function is maximized to obtain θ relative to θ(h+1)To be recycled next time:
(4) (2) are repeated, (3) step obtains θ(h), the sequence of h=1,2 ..., B, abandon before several iterative values, from it is remaining repeatedly
It is averaged in generation value to obtain the estimation of θ
Application effect of the invention is explained in detail below with reference to based on Monte Carlo simulation experiment:
1. simulated conditions
Three different systems of signature, System structural function are considered in simulation study:
System 1 is a 3 component Parallel-series systems, and system signature is (1/3,2/3,0), and structure function is Ψ (X)
=min { X1,max{X2,X3}}。
System 2 is a 4 component Parallel-series systems, and system signature is (1/4,1/4,1/2,0), and structure function is
Ψ (X)=min { X1,max{X2,X3,X4}}。
System 3 is a 4 components mixing parallel system, and system signature is (0,1/2,1/4,1/4), structure function
For Ψ (X)=max { X1,min{X2,X3,X4}}。
Weibull distribution is a widely used service life distribution in survivability modeling because of its flexibility.Enable Y for clothes
The stochastic variable being distributed from Weibull, the then Cumulative Distribution Function of X are as follows:
Because stochastic variable X=lnY obeys minimum the extreme value distribution, minimum the extreme value distribution belongs to Location Scale Families.Therefore,
It is calculated with the logarithm in service life more convenient.The then cumulative distribution function of stochastic variable X=lnY are as follows:
Assuming that the component life of these three systems is distributed Follow Weibull Distribution, logarithm lifetime function obeys minimum point
Cloth, is arranged location parameter μ=1, and scale parameter σ=0.5 carries out analog study.
Carry out the experiment of the big small sample of Monte Carlo through row parameter Estimation to component life using random SEM algorithm,
Simulation process repeats 10000 times, gives the deviation and mean square error, coverage rate, confidence of point estimation based on this 10000 times emulation
Width expected from section, these results are calculated by following formula:
Estimated bias:
Mean square error:
Coverage rate:
Confidence interval mean breadth:Ideal knot
Fruit: estimated bias, mean square error, confidence interval mean breadth are the smaller the better, and coverage rate is then the bigger the better.
2. small sample emulates
For small sample emulate, the present invention consider a life test experiment in have the case where m=10 system, be from
First of thrashing start recording stops to r-th of thrashing, obtains the bilateral censored sample data of an II type.Wherein l
It is determined with r by deleting mistake rate q, such as formula (27), formula (28):
In small sample emulation, because data are less, after deleting mistake rate greater than 50%, utilizable data only have several
A, the deviation of parameter Estimation can greatly increase, it cannot be guaranteed that reasonable estimated accuracy.Therefore in small sample, only consider to delete mistake
Rate is the 50%-0% the case where.By the estimated bias of small sample, mean square error, the coverage rate of confidence interval and desired width meter
It calculates.The operation result of system 1- system 3 such as Fig. 2-Fig. 4.
3. medium to large sample emulation
It being emulated in large sample, the present invention considers there is the case where m=50 system in life test experiment,
It is similar with small sample, it is similarly from first of thrashing start recording, stops to r-th of thrashing, it is double to obtain an II type
Side censored sample data.Wherein l and r is same as above with mistake rate q is deleted, as shown in formula (27) (28).By the medium estimation to large sample
Deviation, mean square error, the coverage rate of confidence interval and desired width calculate.The operation result of system 1- system 3 such as Fig. 5-Fig. 7.
To being analyzed above by the estimated result that random SEM algorithm generates, simulation result diagram result is compared point
It can be found that the result that is showed in random sem emulation of system 1,2,3 is more similar after analysis, it is contemplated that the fact that, will be
(4 component Parallel-series systems, system signature are (1/4,1/4,1/2,0) to system 2, and structure function is Ψ (X)=min { X1,
max{X2,X3,X4Analyzed as representative, and with the previous method using based on maximum likelihood and based on the estimation of recurrence
Method compares.
1. being analyzed for small sample
In terms of estimated bias, present invention discover that the estimated bias that random SEM algorithm obtains all goes to zero, and SEM is to μ and σ
Deviation is negative；In terms of mean square error, random SEM algorithm can be controlled mean square error within 0.05, and previous side
Method can only control deviation within 0.8.Random algorithm is significantly better than previous method for the estimation of μ and σ.In coverage rate side
Face, random SEM algorithm can guarantee reasonable coverage rate；In the mean breadth length of confidence interval, the confidence interval of μ is estimated
Counting range can be within 0.8, can be within 0.5 to the estimation of the confidence interval of σ.
2. for large sample analysis
In terms of estimated bias, the estimated bias of large sample also all goes to zero, and positive and negative deviation and small sample performance are consistent, still
It is negative for random SEM to μ and σ deviation；In terms of mean square error, the mean square error of large sample will be significantly less than small sample, can be with
By mean square error control in the range of 0.005；In terms of coverage rate, the available higher coverage rate of large sample；Confidence interval
Mean breadth on, the estimation of large sample is also smaller with accurate, and the control of the confidence interval of μ, can be with to the confidence interval of σ 0.30
Control is in 0.45.
3. aggregate analysis
In simulations, increase and delete mistake rate the performance of point estimation and interval estimation can be made all to be deteriorated, be embodied in deleting
Mistake rate increases, and variance evaluation MSE increases, and coverage rate CR reduces, and confidence interval WCI becomes longer.
By the comparison of small sample and large sample it can be found that estimated bias in small sample and large sample, deviation is all gradually
It goes to zero, and positive and negative deviation shows unanimously in big small sample, is that deviation is negative.In mean square error, coverage rate and confidence area
Between on, it can be found that the estimation effect of large sample will be much better than small sample.Illustrate that data sample number has played very your writing in estimation
With sample number is more, and estimation is more accurate.Since Small Sample Database is very few in coverage rate, it is greater than 50% when deleting mistake rate, coverage rate
Just it cannot reach zone of reasonableness, therefore when carrying out parameter Estimation to small sample, the mistake rate of deleting of required data cannot be too big.
Identical emulation is carried out with previous estimation method, the simulation result Fig. 8 and Fig. 9 of maximum likelihood (MLE) is based on, is based on back
Simulation result Figure 10, the Figure 11 for returning (REG) are compared with random SEM algorithm and can be found that.Random SEM algorithm is in point estimation
On estimated accuracy be higher than previous Maximum-likelihood estimation and the estimation based on recurrence.And the performance in the estimation in section is simultaneously
It is not above previous method.Therefore random SEM algorithm is more suitable for the point estimation for deleting mistake system bilateral to II type.
The foregoing is merely illustrative of the preferred embodiments of the present invention, is not intended to limit the invention, all in essence of the invention
Made any modifications, equivalent replacements, and improvements etc., should all be included in the protection scope of the present invention within mind and principle.
Claims (10)
1. a kind of component life estimation of distribution parameters method based on random SEM algorithm, which is characterized in that described based on random
The component life estimation of distribution parameters method of SEM algorithm includes:
(1) survival function of lifetime of system is found out with the cumulative distribution function of component life and system signature vector；
(2) parameters of operating part is estimated using random SEM algorithm, S step, with order system signature and conditional probability to
Amount, which combines, obtains life estimate sample, and M step brings estimation sample into likelihood function and maximizes undated parameter；
(3) system small sample and in carrying out Monte Carlo simulation experiment under two kinds of situations of large sample and generate the inclined of point estimation
Difference and width four indices expected from mean square error, coverage rate, confidence interval.
2. as described in claim 1 based on the component life estimation of distribution parameters method of random SEM algorithm, which is characterized in that
The component life estimation of distribution parameters method based on random SEM algorithm specifically includes the following steps:
Step 1, according to System structural function, computing system signature vector SΓ(s1,s2…sn) and order
Signature vectorI-th thrashing is arranged with ω, it is concluded that system
I-th of component of signature vector are as follows:
Event A is that k-th of order thrashing is caused by i-th of order component failure；M are generated to a determining system
Lifetime of system, and the failure for observing which component causes thrashing, repeats this process n times；Then order system
Signature vector can be approximated to be:
Step 2, with the cumulative distribution function F of component lifeXWith system signature vector SΓIndicate the existence letter of lifetime of system
NumberTo n independent same distribution component system, the stochastic variable of component life is indicated with X, lifetime of system is indicated with T
Stochastic variable；The then survival function of lifetime of systemIt indicates are as follows:
Step 3 is established random SEM algorithm and is estimated parameters of operating part；S step, it is general with order system signature and condition
Rate vector combines and obtains life estimate sample；M step brings estimation sample into likelihood function and maximizes undated parameter；Repeat S step
It is walked with M, iteration obtains parameters of operating part estimated value；
Bilateral 100 (1- α) % confidence intervals of step 4, construction confidence interval, component distribution parameter μ and σ can construct as follows:
3. as claimed in claim 2 based on the component life estimation of distribution parameters method of random SEM algorithm, which is characterized in that
The step 3 specifically includes:
(1) right truncation density formula first can be write out according to the sigma-t of order component life and left truncation density formula is distinguished
Are as follows:
(2) for r-l+1 system of first to r-th failure in m system；
(3) likelihood function is maximized to obtain θ relative to θ(h+1)It is recycled next time:
(4) step (2) are repeated, (3) step obtains θ(h), the sequence of h=1,2 ..., B, abandon before several iterative values, from it is remaining repeatedly
It is averaged in generation value to obtain the estimation of θ
4. as claimed in claim 3 based on the component life estimation of distribution parameters method of random SEM algorithm, which is characterized in that
The r-l+1 system for first to r-th failure in m system, component life generate by the following method:
(1) k-th of system (being sorted based on lifetime of system), based on order system signature, with probability mass functionA discrete random variable Λ is generated, this realization is expressed as λ；
(2) it is distributed from the condition of left truncation density, enables θ=θ(h)Generate λ -1 stochastic variables:
(3) it is distributed from the condition of right truncation density, enables θ=θ(h)Generate n- λ stochastic variable:
(4) the pseudo- Complete Sample of system k is finally obtained
(5) whereinTo k=l ..., r repeats step (1)-(3), obtains puppet Complete Sample (r-l+1) n and is to m
Each system of preceding l-1 and rear m-r system in system.
5. as claimed in claim 4 based on the component life estimation of distribution parameters method of random SEM algorithm, which is characterized in that
Described acquisition puppet Complete Sample (r-l+1) n lead to each system of preceding l-1 and rear m-r system in m system, component life
Following method is crossed to generate:
(1) it is based on conditional probability vector p(j:m), with probability mass functionGenerate one
A discrete random variable Δ, it is known that τ=tr:m；
(2) it is distributed from the condition of left truncation density with point of cut-off tk=tr:m, θ=θ(h)Generate δ stochastic variable
(3) it is distributed from right truncation density conditions with point of cut-off tk=tr:m, θ=θ(h)Generate n- δ stochastic variable
(4) (component life) pseudo- Complete Sample of j-th of truncation system is obtained
(5) step (1)-(4) are repeated and obtain estimation sample
6. a kind of component life estimation of distribution parameters side using based on random SEM algorithm described in Claims 1 to 5 any one
The II type of method is bilateral to delete mistake system.
7. a kind of component life estimation of distribution parameters side using based on random SEM algorithm described in Claims 1 to 5 any one
The Vehicle Engineering component life estimation of distribution parameters system of method.
8. a kind of component life estimation of distribution parameters side using based on random SEM algorithm described in Claims 1 to 5 any one
The material engineering component life estimation of distribution parameters system of method.
9. a kind of component life estimation of distribution parameters side using based on random SEM algorithm described in Claims 1 to 5 any one
The electronic engineering component life estimation of distribution parameters system of method.
10. a kind of component life estimation of distribution parameters using based on random SEM algorithm described in Claims 1 to 5 any one
The computer of method.
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| CN102081767A (en) * | 2011-01-29 | 2011-06-01 | 河南科技大学 | Poor information theory fusion-based product life characteristic information extraction method |
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| CN102081767A (en) * | 2011-01-29 | 2011-06-01 | 河南科技大学 | Poor information theory fusion-based product life characteristic information extraction method |
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